Think about how you approach many different types of problems (number sequences included) by taking intuitive 'blind stabs' at a solution. Dissecting that process is more involved than it may at first appear.

>My point is that the program could not learn simply from the correct answers, it would have to learn the processes.

The way I picture such a thing working is that it would somehow produce answers, and work backwards from correct ones to derive the process. In other words, it would be allowed to guess, and be told if the answer was correct or not. One of the things I found intriguing about the approach taken with the program mentioned in the Hofstadter book is that it was NOT given a broad base of knowledge about the rules of mathematics. (!)

Another book I've been reading for about a year now is: Shadows Of The Mind -- by Roger Penrose (which I find to be the mental equivalent of playing follow-the-leader with an olympic decathlon gold-medalist). Penrose examines alternatives to the hypothesis that human thinking is fundamentally algorithmic in nature. He argues that humans are apparently able to somehow arrive at correct solutions to problems that are provably unsolvable by rigidly algorithmic methods.

>BTW, how do you get 13 from "1 1 2 3 5 8"?!

You mean: "What is the process by which you identify the Fibonacci sequence?"